No not redundant, just quite well-known. Look at references for when the cone of curves is not finitely generated. For example, Kovacs' paper 'The cone of curves on a K3 surface' is a nice starting point. In particular, you'll see K3 surfaces with infinitely many $(-2)$-curves. The Fermat quartic $x^4+y^4+z^4+w^4=0$ is one example. Also, if a surface has a large automorphism group and contains one negative curve it typically contains infinitely many - this happens for example for Enriques surfaces, see eg mathoverflow.net/questions/52397/…
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J.C. OttemSep 4 '11 at 18:01

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Some more details: if you blow up the plane at 9 in general position, then acting by Cremona transformations gives a sequence of rational curves of arbitrarily high degree through the 9 points that pull back to divisors of self intersection -1. See Artie Prendergast's note iag.uni-hannover.de/~prendergast/Papers/coneofcurves.pdf for more details.
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J.C. OttemSep 4 '11 at 10:31